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 A056023 Unique triangle such that (1) every positive integer occurs exactly once; (2) row n consists of n consecutive numbers; (3) odd-numbered rows are decreasing; (4) even-numbered rows are increasing; and (5) column 1 is increasing. 7

%I

%S 1,2,3,6,5,4,7,8,9,10,15,14,13,12,11,16,17,18,19,20,21,28,27,26,25,24,

%T 23,22,29,30,31,32,33,34,35,36,45,44,43,42,41,40,39,38,37,46,47,48,49,

%U 50,51,52,53,54,55,66,65,64,63,62,61,60,59,58,57,56

%N Unique triangle such that (1) every positive integer occurs exactly once; (2) row n consists of n consecutive numbers; (3) odd-numbered rows are decreasing; (4) even-numbered rows are increasing; and (5) column 1 is increasing.

%C Self-inverse permutation of the natural numbers.

%C T(2*n-1,1) = A000217(2*n-1) = T(2*n,1) - 1; T(2*n,4*n) = A000217(2*n) = T(2*n+1,4*n+1) - 1. - _Reinhard Zumkeller_, Apr 25 2004

%C Mirror image of triangle in A056011 . [From _Philippe DELEHAM_, Apr 04 2009]

%C Contribution from Clark Kimberling, Feb 03 2011: (Start)

%C When formatted as a rectangle R, for m>1, the numbers n-1

%C and n+1 are neighbors (row, column, or diagonal) of R.

%C R(n,k)=n+(k+n-2)(k+n-1)/2 if n+k is odd;

%C R(n,k)=k+(n+k-2)(n+k-1)/2 if n+k is even.

%C Northwest corner:

%C 1....2....6....7....15...16...28

%C 3....5....8....14...17...27...30

%C 4....9....13...18...26...31...43

%C 10...12...19...25...32...42...49

%C 11...20...24...33...41...50...62

%C (End)

%C a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers. - Boris Putievskiy, Dec 24 2012

%C For generalizations see A218890, A213927. - Boris Putievskiy, Mar 10 2013

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%H Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations Integer Sequences And Pairing Functions</a> arXiv:1212.2732 [math.CO]

%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/PairingFunction.html">MathWorld: Pairng functions</a>

%F T(n, k) = (n^2 - (n - 2*k)*(-1)^(n mod 2)) / 2 + n mod 2. - _Reinhard Zumkeller_, Apr 25 2004

%F a(n)=((i+j-1)*(i+j-2)+((-1)^t+1)*j - ((-1)^t-1)*i)/2, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor[(-1+sqrt(8*n-7))/2]. - Boris Putievskiy, Dec 24 2012

%e Triangle begins : 1 ; 2,3 ; 6,5,4 ; 7,8,9,10 ; 15,14,13,12,11 ; ... [_Philippe DELEHAM_, Apr 04 2009]

%e Enumeration by boustrophedonic ("ox-plowing") diagonal method. - Boris Putievskiy, Dec 24 2012

%t (* As a rectangle: *)

%t r[n_, k_]:=n+(k+n-2)(k+n-1)/2/; OddQ[n+k];

%t r[n_, k_]:=k+(n+k-2)(n+k-1)/2/; EvenQ[n+k];

%t TableForm[Table[f[n, k], {n, 1, 10}, {k, 1, 15}]]

%t Table[r[n-k+1, k], {n, 14}, {k, n, 1, -1}]//Flatten

%t (* Clark Kimberling, Feb 03 2011 *)

%Y Cf. A056011, A218890, A213927.

%K nonn,tabl,easy

%O 1,2

%A _Clark Kimberling_, Aug 01 2000.

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